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Theorem ex-natded5.13 27242
Description: Theorem 5.13 of [Clemente] p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 27243. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
115 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given \$e.
2;32 (𝜓𝜃) (𝜑 → (𝜓𝜃)) Given \$e. adantr 481 to move it into the ND hypothesis
39 𝜏 → ¬ 𝜒) (𝜑 → (¬ 𝜏 → ¬ 𝜒)) Given \$e. ad2antrr 761 to move it into the ND sub-hypothesis
41 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 477
54 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 1,3
65 ... (𝜃𝜏) ((𝜑𝜓) → (𝜃𝜏)) I 5 orcd 407 4
76 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 477
88 ... ...| ¬ 𝜏 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) (sub) ND hypothesis assumption simpr 477
911 ... ... ¬ 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) E 3,8 mpd 15 8,10
107 ... ... 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒) IT 7 adantr 481 6
1112 ... ¬ ¬ 𝜏 ((𝜑𝜒) → ¬ ¬ 𝜏) ¬I 8,9,10 pm2.65da 599 7,11
1213 ... 𝜏 ((𝜑𝜒) → 𝜏) ¬E 11 notnotrd 128 12
1314 ... (𝜃𝜏) ((𝜑𝜒) → (𝜃𝜏)) I 12 olcd 408 13
1416 (𝜃𝜏) (𝜑 → (𝜃𝜏)) E 1,6,13 mpjaodan 826 5,14,15

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 481; simpr 477 is useful when you want to depend directly on the new assumption). (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses
Ref Expression
ex-natded5.13.1 (𝜑 → (𝜓𝜒))
ex-natded5.13.2 (𝜑 → (𝜓𝜃))
ex-natded5.13.3 (𝜑 → (¬ 𝜏 → ¬ 𝜒))
Assertion
Ref Expression
ex-natded5.13 (𝜑 → (𝜃𝜏))

Proof of Theorem ex-natded5.13
StepHypRef Expression
1 simpr 477 . . . 4 ((𝜑𝜓) → 𝜓)
2 ex-natded5.13.2 . . . . 5 (𝜑 → (𝜓𝜃))
32adantr 481 . . . 4 ((𝜑𝜓) → (𝜓𝜃))
41, 3mpd 15 . . 3 ((𝜑𝜓) → 𝜃)
54orcd 407 . 2 ((𝜑𝜓) → (𝜃𝜏))
6 simpr 477 . . . . . 6 ((𝜑𝜒) → 𝜒)
76adantr 481 . . . . 5 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒)
8 simpr 477 . . . . . 6 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)
9 ex-natded5.13.3 . . . . . . 7 (𝜑 → (¬ 𝜏 → ¬ 𝜒))
109ad2antrr 761 . . . . . 6 (((𝜑𝜒) ∧ ¬ 𝜏) → (¬ 𝜏 → ¬ 𝜒))
118, 10mpd 15 . . . . 5 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)
127, 11pm2.65da 599 . . . 4 ((𝜑𝜒) → ¬ ¬ 𝜏)
1312notnotrd 128 . . 3 ((𝜑𝜒) → 𝜏)
1413olcd 408 . 2 ((𝜑𝜒) → (𝜃𝜏))
15 ex-natded5.13.1 . 2 (𝜑 → (𝜓𝜒))
165, 14, 15mpjaodan 826 1 (𝜑 → (𝜃𝜏))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386 This theorem is referenced by: (None)
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